Rock didn't really push Big Band or Jazz aside in the 50's. For starters, Big Band was already a dying art form that saw it's best days before WW2. There were still some major Big Bands out there after WW2, like the Dorseys and Stan Kenton, but the Big Bands mostly retired from touring by the 50's, and the few holdouts were playing clubs in one town. The main reason for this was money. A Big Band is incredibly expensive to keep on the road. Big Bands usually have 10-15 musicians who all require transportation to the gig, meals, hotel accommodation, and a salary. The window of opportunity for Big Band Jazz was smaller than most people realize. Before the 20's Barbershop Quartets were the biggest pop acts (yeah, really, barbershop quartets), and in the 20's vaudeville/Broadway style music was on top. The era of Big Bands started in the teens and 20's, and ran through the 60's and 70's when Don Ellis and Maynard Ferguson had major albums, but the stretch of Big Band as a consistent chart topper was really only from the 30's till WW2.

After WW2 Jazz stopped being about Big Bands because no one could afford to keep a big band on the road, so small ensembles took over that niche. Jazz didn't just switch instrumentation though, it switched style pretty dramatically in the 50's. Jazz performers of that era were intensely creative and introspective, and stopped playing songs people could dance to and started focusing on pieces that featured long improvisational soloing and complex rhythms. Jazz walked away from a pop dance chart mentality and left that niche in the market pretty much wide open.

At the same time the top vocalists were crooners. The best of them were so popular they could afford to keep a large Big Band on the payroll just for backup duty. Bing Crosby, who is arguably the biggest "star" produced by the American entertainment industry in the 20th century is an example of a crooner. Frank Sinatra, Nat King Cole and Perry Como also fall into this type. These guys are what the returning soldiers from WW2 listened too. They sold albums by the truckload, but again, they were producing ballads, not upbeat pop dance tunes.

That is where rock came in. Rock was appealing to the audience Jazz, with it's introspective to the point of being masturbatory soloing, and crooners, with their super smooth slow paced ballads were ignoring. Some of the early attempts to describe rock considered it a new take on swing music, swing of course being a major dance craze jazz produced in the 30's. Rock was dance music first and foremost in the early days. It's no coincidence that Elvis rose to fame and notoriety on the basis of his dancing, with the music almost considered an afterthought.

The thing is, people didn't stop buying Frank Sinatra albums one day and start buying Elvis, Chuck Berry and Little Richard the next. People who bought Sinatra and Perry Como albums in the 40's were still buying Sinatra and Perry Como albums in the 60's and 70's, and these crooners were still selling out shows and charting albums. Young people entering the market in the 50's started buying the rock and R&B albums they liked dancing to and they kept buying more of that stuff. There was no light switch type moment where one type of music replaced the other, or even an organic it used to be this now it's this happening over time. Older listeners had one style, and younger listeners had their style, and gradually as more young people bought more albums the music they preferred developed into what we now think of as a sort of mass movement. You could still buy jazz and crooner albums in record stores, and listen to them on the radio throughout all of this.

There were other forms of music going on at the same time too. The music industry was seriously shocked in the 90's when they finally learned how to track actual album sales instead of estimates based on faulty models and discovered that country music and gospel were more popular than rap and rock. Rock never really took over the radio airwaves, it carved out a niche on them next to old fashioned (by this time) pop, jazz, classical, and country.